Time domain measuring method with calibration in the frequency range

ABSTRACT

A method for determining an electric voltage u(t) and/or an electric current i(t) of an HF signal in an electrical cable on a calibration plane by measuring in the time domain. Using a directional coupler, a first portion v 3 (t) of a first HF signal is decoupled, fed to a time domain measuring device, and a second portion v 4 (t) of a second HF signal is decoupled. The signal portions v 3 (t), v 4 (t) are converted into the frequency domain, then absolute wave frequencies in the frequency domain are determined and converted into the electric voltage u(t) and/or the electric current i(t). In a previous calibration step, the calibration parameters are determined, and the absolute wave frequencies on the calibration plane are determined using the calibration parameters (e 00,r (Γ 3 , Γ 4 ), e 01,r (Γ 3 , Γ 4 ), e 10,r (Γ 3 , Γ 4 ), e 11,r (Γ 3 , Γ 4 )), wherein Γ 3 , Γ 4  are the reflection factors of the inputs of the time domain measuring device.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to a method for determining an electric voltageu(t) and/or an electric current i(t) of a RF signal on an electric cablein a calibration plane through measurement in the time domain using atime domain measuring device, whereby the calibration plane is designedsuch that a device under test can be connected electrically with thecalibration plane. In a measuring step, using a directional coupler, afirst component v₃(t) of a first RF signal which, starting out from asignal input, runs in the direction of the calibration plane through thedirectional coupler, is decoupled, fed to the time domain measuringdevice at a first measuring input and measured there in a firstmeasuring plane, and a second component v₄(t) of a second RF signalwhich, starting out from the calibration plane, runs in the direction ofthe signal input through the directional coupler, is also decoupledusing the directional coupler, fed to the time domain measuring deviceat a second measuring input and measured there in a second measuringplane. The signal components v₃(t), v₄(t) measured using the time domainmeasuring device are, by means of a first mathematical operation,transformed into the frequency domain as wave quantities V₃(f) andV₄(f), then absolute wave quantities a₂ and b₂ in the frequency domainare determined in the calibration plane from the wave quantities V₃(f)and V₄(f) using calibration parameters, and fmally the calculatedabsolute wave quantities a₂ and b₂ are, by means of a secondmathematical operation, converted into the electric voltage u(t) and/orthe electric current i(t) of the RF signal in the time domain in thecalibration plane. In a preceding calibration step, the calibrationparameters are determined in such a way that they link the wavequantities V₃(f) and V₄(f) in the measuring planes mathematically withthe wave quantities a₂ and b₂ in the calibration plane.

2. Description of Related Art

One of the most important measuring tasks in radio frequency andmicrowave technology involves the measurement of reflection coefficientsor generally—in the case of multiports—the measurement of scatteringparameters. The linearly-describable network behavior of a device undertest (DUT) is characterized through the scattering parameters.Frequently, it is not only the scattering parameters at a singlemeasuring frequency which are of interest, but theirfrequency-dependency over a finitely broad measuring bandwidth. Theassociated measuring method is referred to as network analysis.Depending on the importance of the phase information in the measuringtask in question, the scattering parameters can either be measuredsolely in terms of amount or also as a complex measurement. In the firstcase one speaks of scalar network analysis, in the second case ofvectorial network analysis. Depending on the method, number of ports andmeasuring frequency range, the network analyzer is a more or lesscomplex system consisting of test signal source and receivers whichfunction according to the homodyne or the heterodyne principle. Becausethe measuring signals have to be fed to the device under test and backagain through cables and other components with unknown and non-idealproperties, in addition to random errors, system errors also occur innetwork analysis. Through calibration measurements, the aim of which isto determine as many as possible of the unknown parameters of the testapparatus, the system errors can, within certain limits, be reversed.Very many methods and strategies exist here which differ considerably inthe scope of the error model and thus in complexity and efficiency. (UweSiart; “Calibration of Network Analysers”; 4 Jan. 2012 (Version 1.51);http://www.siart.de/lehre/nwa.pdf).

However, scattering parameters measured in such a calibrated manner onlyfully describe linear, time-invariant devices under test. The Xparameters represent an expansion of the scattering parameters tonon-linear devices under test (D. Root, et al: “X-Parameters: The newparadigm for describing non-linear RF and microwave components.” In:tm—Technisches Messen No. 7-8, Vol. 77, 2010), which are also definedthrough the frequency. However, each device under test can also bedescribed through measurement of the currents and voltages or theabsolute wave quantities at its ports within the time domain. Themeasurement in the time domain inherently includes all spectralcomponents resulting for example from the non-linearity as well as thechange over time of the device under test or its input signals. Such atime domain measurement also requires calibration. However, in order tomeasure absolute values the aforementioned calibration methods cannot beapplied without modification, since they only permit the determinationof relative values (scattering parameters).

A high frequency circuit analyzer which is used to test amplifiercircuits is known from WO2003/048791 A2. A microwave transition analyzer(MTA) with two inputs measures two independent signal waveforms, forexample the propagated and reflected wave, via signal paths and ports inthe time domain while the amplifier circuit under test is connected. Themeasured waves are further processed by means of calibration data inorder to compensate for the influence of the measurement system on thewaves between the ports of the amplifier circuit and the input ports ofthe MTA. The MTA is again used in order to determine the calibrationdata, measuring signals in the time domain while the calibrationstandards are connected. These signals in the time domain are convertedinto the frequency domain using an FFT and the calibration data are thendetermined. Since only periodic signals in the time domain are measured,the signals are converted to a lower-frequency intermediate frequencyprior to measurement.

The document WO2013/143650 A1 describes a time domain measuring methodwith calibration in the frequency domain according to the preamble ofclaim 1. In this method, an electric voltage and/or an electric currentof a high frequency signal are measured in the time domain on anelectric conductor in a calibration plane. For this purpose, adirectional coupler is inserted in the line supplying the measurementsignal to the device under test, and a first component of the first HFsignal, which runs from the signal input of the directional couplerthrough the directional coupler in the direction of the device undertest, is decoupled via the first measuring output of the directionalcoupler and measured using the time domain measuring device, and asecond component of the HF signal returning from the device under test,which runs in the opposite direction through the directional coupler, isdecoupled via the second measuring output of the directional coupler andmeasured using the time domain measuring device. The measured signalcomponents are transformed into the frequency domain in order to obtainwave quantities. With the aid of previously determined calibrationparameters, corresponding wave quantities in the calibration plane aredetermined in the frequency domain from these wave quantities determinedin the measuring planes, and these wave quantities are then in turntransformed back into the time domain, so that they state the signalvalues u(t) and/or i(t) in the time domain which are to be determined inthe calibration plane.

The calibration parameters which link the wave quantities in themeasuring planes with the wave quantities in the calibration plane are,in a preceding calibration step, determined in a frequency-dependentmanner with the aid of a calibration device, whereby the calibrationstep is described in detail in the cited document WO2013/143650 A1.These calibration parameters can be represented in the form of an errormatrix

${E==\begin{pmatrix}e_{00} & e_{01} \\e_{10} & e_{11}\end{pmatrix}},$

with which the wave quantities a₂, b₂ in the calibration plane can becalculated as follows from wave quantities b₄, b₃ in the measuringplanes:

$\begin{pmatrix}b_{4} \\b_{2}\end{pmatrix} = {\begin{pmatrix}e_{00} & e_{01} \\e_{10} & e_{11}\end{pmatrix}{\begin{pmatrix}b_{3} \\a_{2}\end{pmatrix}.}}$

The disclosing content of WO2013/143650 A1 is, with respect to thedetermination of the calibration parameters, herewith included in thisdescription through express reference.

However, it has transpired that the signal values in the calibrationplane determined by means of this method are not always exact and candepend on the time domain measuring device used.

Known from WO-A-2013 143 650 is a method for determining an electricvoltage and/or an electric current of an RF signal on an electricconductor in a calibration plane through measurement in the time domainusing a time domain measuring device, wherein a device under test can beconnected electrically with the calibration plane. In a measuring step,using a directional coupler, a first component of a first RF signalwhich, starting out from a signal input, runs in the direction of thecalibration plane through the directional coupler, is decoupled, fed tothe time domain measuring device at a first measuring input and measuredthere. A second component of a second RF signal which, starting out fromthe calibration plane, runs in the direction of the signal input throughthe directional coupler, is fed to the time domain measuring device at asecond measuring input and measured there. The signal components are, bymeans of a first mathematical operation, transformed into the frequencydomain as wave quantities, then from these wave quantities absolute wavequantities in the frequency domain are determined in the calibrationplane using calibration parameters, and finally the determined absolutewave quantities are, by means of a second mathematical operation,converted into the electric voltage and/or the electric current of theRF signal in the time domain in the calibration plane, wherein thecalibration parameters link the wave quantities mathematically with theabsolute wave quantities in the calibration plane.

The document Clement T. S. et al: “Calibration of Sampling OscilloscopesWith High-Speed Photodiodes,” IEEE TRANSACTIONS ON MICROWAVE THEORY ANDTECHNIQUES, IEEE SERVICE CENTER, PISCATAWAY, N.J., US, vol. 54, no. 8, 1Aug. 2006 (2006-08-01), pages 3173-3181, XP-001545193, ISSN: 0018-9480,DOI: 10.1109/TMTT.2006.879135 section: C. Impedance Mismatch Correctiondiscloses the determination of reflection coefficients of a photodiodeand an oscilloscope with the aid of a network analyzer.

The document Arkadiusz Lewandowski et al: “Covariance-BasedVector-Network-Analyzer Uncertainty Analysis for Time andFrequency-Domain Measurements”, IEEE TRANSACTIONS ON MICROWAVE THEORYAND TECHNIQUES, IEEE SERVICE CENTER, PISCATAWAY, N.J., US, vol. 58, no.7, 1 Jul. 2010 (2010-07-01), pages 1877-1886, XP-011311287, ISSN:0018-9480, section: IV. Propagating Covariance-Matrix-BasedUncertainties subsection; A Mismatched-Correcting Waveform Measurementsdiscloses the determination of an output impedance of the signal sourceand an input impedance of the oscilloscope using a network analyser,whereby the equation used is only valid for low frequencies, but not forhigh frequencies.

The document WO-A-2008 016699 discloses the determination of variousparameters for various components such as test prods or cables.

SUMMARY OF THE INVENTION

In view of this problem, the invention is based on the problem ofproviding an improved measuring method for high frequency currents andvoltages and absolute wave quantities in the time domain.

This problem is solved according to the invention through a furtherdevelopment of the method described above, which is substantiallycharacterized in that the first measuring input of the time domainmeasuring device has a known (complex-valued) reflection coefficientΓ₃≠0 and/or the second measuring input of the time domain measuringdevice has a known (complex-valued) reflection coefficient Γ₄≠0,wherein, in the calibration step, the calibration parameters e_(00,r),e_(01,r), e_(10,r), e_(11,r) are determined with the aid of acalibration device in relation to the frequency f and in relation to thereflection coefficient at at least one of the measuring inputs of thetime domain measuring device, and in the measuring step the wavequantities a₂ and b₂ are determined from the wave quantities V₃(f) andV₄(f) using the calibration parameters e_(00,r)(Γ₃, Γ₄), e_(01,r)(Γ₃,Γ₄), e_(10,r)(Γ₃, Γ₄), e_(11,r)(Γ₃, Γ₄).

The above and other objects, which will be apparent to those skilled inthe art, are achieved in the present invention which is directed to amethod for determining an electric voltage u(t) and/or an electriccurrent i(t) of a RF signal on an electric cable in a calibration planethrough measurement in the time domain using a time domain measuringdevice, wherein a device under test can be connected electrically withthe calibration plane, wherein, in a measuring step, using a directionalcoupler, a first component v₃(t) of a first RF signal which, startingout from a signal input, runs in the direction of the calibration planethrough the directional coupler, is decoupled, fed to the time domainmeasuring device at a first measuring input and measured there, and asecond component v₄(t) of a second RF signal which, starting out fromthe calibration plane, runs in the direction of the signal input throughthe directional coupler, is decoupled, fed to the time domain measuringdevice at a second measuring input and measured there, wherein thesignal components v₃(t), v₄(t) are, by a first mathematical operation,transformed into the frequency domain as wave quantities V₃(f) andV₄(f), then absolute wave quantities a₂ and b₂ in the frequency domainare determined in the calibration plane from the wave quantities V₃(f)and V₄(f) using calibration parameters (e_(00,r), e_(01,r), e_(10,r),e_(11,r)), and the determined absolute wave quantities a₂ and b₂ are, bya second mathematical operation, converted into the electric voltageu(t) and/or the electric current i(t) of the RF signal in the timedomain in the calibration plane, wherein the calibration parameters linkthe wave quantities V₃(f) and V₄(f) mathematically with the absolutewave quantities a₂ and b₂ in the calibration plane, such that the firstmeasuring input of the time domain measuring device has a reflectioncoefficient Γ₃≠0 and/or the second measuring input of the time domainmeasuring device has a reflection coefficient Γ₄≠0, and in a precedingcalibration step, the calibration parameters (e_(00,r), e_(01,r),e_(10,r), e_(11,r)) are determined, with the aid of a calibrationdevice, in relation to the frequency f and in relation to a calibrationstandard with known reflection coefficient Γ_(DUT) of at least one ofthe measuring inputs of the time domain measuring device, and the wavequantities a₂ and b₂ are determined in the measuring step from the wavequantities V₃(f) and V₄(f) using the calibration parameters(e_(00,r)(Γ₃, Γ₄), e_(01,r)(Γ₃, Γ₄), e_(10,r)(Γ₃, Γ₄), e_(11,r)(Γ₃,Γ₄)), wherein, during the calibration step the signal input of thedirectional coupler is connected with a first measuring port S1, thefirst measuring output of the directional coupler is connected with asecond measuring port S3 and the second measuring output of thedirectional coupler is connected with a third measuring port S4 of thecalibration device, wherein one or more measuring standards with knownreflection coefficients are connected to a signal output of thedirectional coupler connected with the calibration plane S2, wherein thecalibration parameters (e_(00,r), e_(01,r), e_(10,r,) e_(11,r)) link thewave quantity b₃ running in at the second measuring port S3 and the wavequantity b₄ running in at the third measuring port S4 with the wavequantities b₂, a₂ running in and out in the calibration plane (14, S2)as follows:

$\begin{pmatrix}b_{4} \\b_{2}\end{pmatrix} = {\begin{pmatrix}e_{00,r} & e_{01,r} \\e_{10,r} & e_{11,r}\end{pmatrix}\begin{pmatrix}b_{3} \\a_{2}\end{pmatrix}}$

wherein the scattering parameters S_(xy) (x=1-4, y=1-4) of thescattering matrix S of the four-port with the ports S1, S2, S3, S4, inparticular of the directional coupler together with input cables, aredetermined with the aid of the calibration apparatus, wherein thecalibration parameters e_(00,r), e_(01,r), e_(10,r), e_(11,r), inrelation to the reflection coefficients of the time domain measuringdevice Γ₃ Γ₄ are determined from the scattering parameters S_(xy),wherein the calibration parameters are determined from the scatteringparameters as follows:

$\mspace{20mu} {{e_{00,r} = \frac{S_{41} - {\Gamma_{3}S_{33}S_{41}} + {\Gamma_{3}S_{31}S_{43}}}{S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}}},\mspace{20mu} {e_{01,r} = \frac{{S_{31}S_{42}} - {S_{32}S_{41}}}{S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}}},{e_{10,r} = {\frac{{\Gamma_{4}S_{24}S_{41}} + {\Gamma_{3}\begin{pmatrix}{{\Gamma_{4}{S_{24}\left( {{S_{31}S_{43}} - {S_{33}S_{41}}} \right)}} +} \\{S_{23}\left( {S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}} \right)}\end{pmatrix}}}{S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}} + \frac{S_{21}\left( {1 - {\Gamma_{4}S_{44}} - {\Gamma_{3}\left( {S_{33} + {\Gamma_{4}S_{34}S_{43}} - {\Gamma_{4}S_{33}S_{44}}} \right)}} \right)}{S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44\;}}}}},{e_{11,r} = {\frac{\begin{matrix}{{\Gamma_{4}{S_{24}\left( {{{- S_{32}}S_{41}} + {S_{31}S_{42}}} \right)}} + {S_{22}\left( {S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}} \right)} -} \\{S_{21}\left( {S_{32} + {\Gamma_{4}S_{34}S_{42}} - {\Gamma_{4}S_{32}S_{44}}} \right)}\end{matrix}}{S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}}.}}}$

wherein the scattering parameters S_(xy) are determined throughmeasurement of the values b₁/a₁, b₃/a₃, b₄/a₄, b₃/a₁ or b₁/a₃, b₄/a₁ orb₁/a₄, b₄/a₃ or b₃/a₄ at the measuring ports S1, S3, S4 of thecalibration device, wherein in each case preferably the measuringstandards Match (M), Open (O), Short (S) with the known reflectioncoefficients Γ_(M), Γ_(O), Γ_(S) are connected as devices under test inthe calibration plane S2, where a₁, a₃, a₄ are wave quantities runningin at the respective measuring ports S1, S3, S4 and b₁, b₃, b₄ are wavequantities running out at the respective measuring ports S1, S3, S4, andwherein the scattering parameters S_(xy) are determined by means of thefollowing equations:

S₁₁ = i₀₀ S₂₁ = i₁₀ S₁₂ = i₀₁ S₂₂ = i₁₁${S_{13} = {S_{31} = {{\overset{\sim}{S}}_{31} - {\frac{\Gamma_{DUT}i_{10}}{1 - {\Gamma_{DUT}i_{11}}}S_{32}}}}},{S_{14} = {S_{41} = {{{\overset{\sim}{S}}_{41} - {\frac{\Gamma_{DUT}i_{10}}{1 - {\Gamma_{DUT}i_{11}}}{S_{42}.S_{23}}}} = {S_{32} = \frac{{- \left( {e_{11} - i_{11}} \right)}\left( {{\Gamma_{DUT}i_{11}} - 1} \right){\overset{\sim}{S}}_{31}}{\left( {{e_{11}\Gamma_{DUT}} - 1} \right)i_{10}}}}}},{S_{24} = {S_{42} = \frac{\left( {{\Gamma_{DUT}i_{11}} - 1} \right)\left( {{e_{01}i_{10}} + {\left( {i_{11} - e_{11}} \right){\overset{\sim}{S}}_{41}}} \right)}{\left( {{e_{11}\Gamma_{DUT}} - 1} \right)i_{10}}}},{S_{33} = {{{\overset{\sim}{S}}_{33} - {\frac{\Gamma_{DUT}S_{23}}{1 - {\Gamma_{DUT}i_{111}}} \cdot {S_{32}.S_{43}}}} = {S_{34} = {{{\overset{\sim}{S}}_{34} - {\frac{\Gamma_{DUT}S_{24}}{1 - {\Gamma_{DUT}i_{11}}} \cdot {S_{32}.S_{44}}}} = {{\overset{\sim}{S}}_{44} - {\frac{\Gamma_{DUT}S_{24}}{1 - {\Gamma_{DUT}i_{11}}} \cdot S_{42}}}}}}},$

where:Γ_(DUT) is the known reflection coefficient of the calibration standardused during the measurement;

are the b_(x)/a_(y) measurable at the measuring ports S1, S3, S4; and

${i_{00} = {\overset{\sim}{S}}_{11,M}},{e_{00} = \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}},{{i_{10}i_{01}} = \frac{\left( {\Gamma_{O} - \Gamma_{S}} \right)\left( {{\overset{\sim}{S}}_{11,O} - {\overset{\sim}{S}}_{11,M}} \right)\left( {{\overset{\sim}{S}}_{11,S} - {\overset{\sim}{S}}_{11,M}} \right)}{\Gamma_{O}{\Gamma_{S}\left( {{\overset{\sim}{S}}_{11,O} - {\overset{\sim}{S}}_{11,S}} \right)}}},{{e_{10}e_{01}} = \frac{\left( {\Gamma_{O} - \Gamma_{S}} \right)\left( {\frac{{\overset{\sim}{S}}_{41,O}}{{\overset{\sim}{S}}_{31,O}} - \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}} \right)\left( {\frac{{\overset{\sim}{S}}_{41,S}}{{\overset{\sim}{S}}_{31,S}} - \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{21,M}}} \right)}{\Gamma_{O}{\Gamma_{S}\left( {\frac{{\overset{\sim}{S}}_{41,O}}{{\overset{\sim}{S}}_{31,O}} - \frac{{\overset{\sim}{S}}_{41,S}}{{\overset{\sim}{S}}_{31S}}} \right)}}},{i_{11} = \frac{{\Gamma_{S}\left( {{\overset{\sim}{S}}_{11,O} - {\overset{\sim}{S}}_{11,M}} \right)} - {\Gamma_{O}\left( {{\overset{\sim}{S}}_{11,S} - {\overset{\sim}{S}}_{11,M}} \right)}}{\Gamma_{O}{\Gamma_{S}\left( {{\overset{\sim}{S}}_{11,O} - {\overset{\sim}{S}}_{11,S}} \right)}}},{e_{11} = \frac{{\Gamma_{S}\left( {\frac{{\overset{\sim}{S}}_{41,O}}{{\overset{\sim}{S}}_{31,O}} - \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}} \right)} - {\Gamma_{O}\left( {\frac{{\overset{\sim}{S}}_{41,S}}{{\overset{\sim}{S}}_{31,S}} - \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}} \right)}}{\Gamma_{O}{\Gamma_{S}\left( {\frac{{\overset{\sim}{S}}_{41,O}}{{\overset{\sim}{S}}_{31,O}} - \frac{{\overset{\sim}{S}}_{41,S}}{{\overset{\sim}{S}}_{31,S}}} \right)}}},$

where Γ_(O), Γ_(S), Γ_(M) are known reflection coefficients of thecalibration standards Open (O), Short (S) and Match (M), and

are the b_(x)/a_(y) measurable at the measuring ports with connectedcalibration standard K.

BRIEF DESCRIPTION OF THE DRAWINGS

The features of the invention believed to be novel and the elementscharacteristic of the invention are set forth with particularity in theappended claims. The figures are for illustration purposes only and arenot drawn to scale. The invention itself, however, both as toorganization and method of operation, may best be understood byreference to the detailed description which follows taken in conjunctionwith the accompanying drawings in which:

FIG. 1 shows a schematic representation of a measuring setup forcarrying out a calibration step of the method according to the inventionin the frequency domain;

FIG. 2 shows a schematic representation of a measuring setup forcarrying out a measuring step of the method according to the inventionin the time domain;

FIG. 3 shows a signal flow diagram of the error two-port with the errormatrix E between measuring outputs b₃, b₄ of the directional coupler andthe calibration plane (FIG. 3a ) and a signal flow diagram of the errortwo-port I between the signal input of the directional coupler and thecalibration plane (FIG. 3b ) for the measuring setup according to FIG.1;

FIG. 4 shows a representation of the four-port with the scatteringmatrix S (directional coupler together with input cables, FIG. 4a ), arepresentation of the error two-port with the error matrix E (FIG. 4b )and a representation of the error two-port I (FIG. 4c ) for themeasuring setup according to FIG. 1;

FIG. 5 shows amounts of the entries S_(xy) (x=1-4, y=1-4) of thescattering matrix S of an exemplary setup in relation to the frequency(x-axis: frequency f/Hz; y-axis: /S_(xy)//dB); the continuous lines showthe values determined in the calibration step; the crosses show asimulated reference;

FIG. 6a shows a graphic representation of entries e₀₀, e₀₁, e₁₀ and e₁₁of the error matrix E determined in the calibration step of the methodaccording to the invention for the exemplary setup shown in FIG. 5, inrelation to a frequency f;

FIG. 6b shows a graphic representation of the calibration parameterse_(00,r)(Γ₃, Γ₄), e_(01,r)(Γ₃, Γ₄), e_(10,r)(Γ₃, Γ₄), e_(11,r)(Γ₃, Γ₄)for the exemplary setup shown in FIG. 5, in relation to the frequency f,using exemplary reflection coefficients Γ₃, Γ₄ of the measuring inputsof the time domain measuring device;

FIG. 7a shows a graphic representation of an electric voltage u(t)determined in the calibration plane with the method according to theinvention using the (corrected) calibration parameters e_(00,r),e_(01,r), e_(10,r), e_(11,r) and using the (uncorrected) calibrationparameters e₀₀, e₀₁, e₁₀, e₁₁ for an input first RF signal; and

FIG. 7b shows a graphic representation of an electric current i(t)determined in the calibration plane with the method according to theinvention using the (corrected) calibration parameters e_(00,r),e_(01,r), e_(10,r), e_(11,r) and using the (uncorrected) calibrationparameters e₀₀, e₀₁, e₁₀, e₁₁ for an input first RF signal.

DESCRIPTION OF THE PREFERRED EMBODIMENT(S)

In describing the preferred embodiment of the present invention,reference will be made herein to FIGS. 1-7 of the drawings in which likenumerals refer to like features of the invention.

The invention is based on the knowledge that the method described in thedocument WO2013/143650 A1 only provides exact results if the twomeasuring inputs of the time domain measuring device havereflection-free terminations. When carrying out the calibration it wasassumed that the calibration apparatus had already been calibratedpreviously to the measuring ports used and will thus behave ideally. Onetherefore obtains error matrices E and I for an ideal adaptation of themeasuring ports of the calibration apparatus. In contrast, thecalibration parameters which are necessary during the measuring step inorder to obtain an exact measurement depend on the reflectioncoefficients Γ₃ and Γ₄ at the measuring inputs of the time domainmeasuring device. The conventional error matrix E thus only leads toexact results if, for the measuring inputs, it is the case that Γ₃=Γ₄=0.

In contrast, according to the invention any time domain measuringdevices can be connected to the measuring outputs of the directionalcoupler for the measurement in the time domain, since the calibrationparameters are determined in relation to the reflection coefficients atthe measuring inputs of the time domain measuring device. The reflectioncoefficients Γ₃ and Γ₄ of the measuring inputs of the time domainmeasuring device are known when determining the voltage and/or thecurrent in the measuring step, or can be determined through separatemeasurement, so that the frequency-dependent e_(00,r)(Γ₃, Γ₄),e_(01,r)(Γ₃, Γ₄), e_(10,r)(Γ₃, Γ₄), e_(11,r)(Γ₃, Γ₄) can be used ascalibration parameters.

In the calibration step, it has proved practical to connect the signalinput of the directional coupler with a first measuring port S1, toconnect the first measuring output of the directional coupler with asecond measuring port S3 and the second measuring output of thedirectional coupler with a third measuring port S4 of the calibrationdevice. At the same time, one or more measuring standards with knownreflection coefficients Γ_(k) are connected to a signal output of thedirectional coupler connected with the calibration plane S2.

The determination of the wave quantities a₂ and b₂ in the calibrationplane from the wave quantities b₃ and b₄ in the measuring planes can becarried out particularly quickly and reliably if the calibrationparameters e_(00,r), e_(01,r), e_(10,r), e_(11,r) link the wave quantityb₃ running in at the second measuring port S3 and the wave quantity b₄running in at the third measuring port S4 with the wave quantities b₂,a₂ running in and out in the calibration plane S2 as follows:

$\begin{pmatrix}b_{4} \\b_{2}\end{pmatrix} = {\begin{pmatrix}e_{00,r} & e_{01,r} \\e_{10,r} & e_{11,r}\end{pmatrix}{\begin{pmatrix}b_{3} \\a_{2}\end{pmatrix}.}}$

According to the invention, the determination of the calibrationparameters e_(00,r), e_(01,r), e_(10,r), e_(11,r) can be simplified inthat the scattering parameters S_(xy) (x=1-4, y=1-4) of the scatteringmatrix S of the four-port with the ports S1, S2, S3, S4, in particularthe scattering matrix S of the directional coupler together with inputcables, are determined with the aid of the calibration apparatus, andthe calibration parameters e_(00,r), e_(01,r), e_(10,r), e_(11,r) inrelation to the reflection coefficients of the time domain measuringdevice Γ₃ and Γ₄ are then determined from the scattering parametersS_(xy).

In other words, the four-port whose scattering parameters S_(xy) aredetermined has the following four ports:

-   -   a first port S1 which represents the signal input of the        directional coupler and which is connected with the first        measuring port of the calibration apparatus during calibration,    -   a second port S2, which is connected with the signal output of        the directional coupler and represents the calibration plane,        whereby the devices under test and the measuring standards can        be connected at the second port S2,    -   a third port S3, which represents the first signal output of the        directional coupler or is connected with this, and which is        connected with the second measuring port of the calibration        apparatus during calibration,    -   a fourth port S4, which represents the second signal output of        the directional coupler or is connected with this, and which is        connected with the third measuring port of the calibration        apparatus during calibration.

This four-port is present in unchanged form during the calibration stepand during the measuring step (or changes in the input cable to thedirectional coupler have no effect, since the terms dependent on theinput cable are eliminated during the determination of e_(xy)), so thatthe determined scattering parameters S_(xy) of the four-port determinedduring the calibration are still correct during the measuring step andcan be used for determination of the corrected error matrix E_(r). Inthis relationship it is to be emphasised that the scattering parametersof an n-port are by definition independent of the external wiring. Incontrast, the error matrix E_(r), the terms of which are needed duringthe measurement, depends on the reflection coefficients at the measuringports of the time domain measuring device, but not on the properties ofthe input cable between the signal generator and the signal input of thedirectional coupler.

In other words, the (uncorrected) error matrix E, as determined in thedocument WO2013/143650 A1, is only valid where Γ₃=Γ₄=0, which is notgenerally guaranteed in a time domain measuring device. This can affectthe measuring accuracy of the method described in the documentWO2013/143650 A1.

However, with the method according to the invention, a corrected errormatrix E_(r) can be determined from the scattering parameters S_(xy)using the known reflection coefficients Γ₃, Γ₄ of the time domainmeasuring device, as follows:

$\mspace{20mu} {{e_{00,r} = \frac{S_{41} - {\Gamma_{3}S_{33}S_{41}} + {\Gamma_{3}S_{31}S_{43}}}{S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}}},\mspace{20mu} {e_{01,r} = \frac{{S_{31}S_{42}} - {S_{32}S_{41}}}{S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}}},{e_{10,r} = {\frac{{\Gamma_{4}S_{24}S_{41}} + {\Gamma_{3}\begin{pmatrix}{{\Gamma_{4}{S_{24}\left( {{S_{31}S_{43}} - {S_{33}S_{41}}} \right)}} +} \\{S_{23}\left( {S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}} \right)}\end{pmatrix}}}{S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}} + \frac{S_{21}\left( {1 - {\Gamma_{4}S_{44}} - {\Gamma_{3}\left( {S_{33} + {\Gamma_{4}S_{34}S_{43}} - {\Gamma_{4}S_{33}S_{44}}} \right)}} \right)}{S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44\;}}}}},{e_{11,r} = {\frac{\begin{matrix}{{\Gamma_{4}{S_{24}\left( {{{- S_{32}}S_{41}} + {S_{31}S_{42}}} \right)}} + {S_{22}\left( {S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}} \right)} -} \\{S_{21}\left( {S_{32} + {\Gamma_{4}S_{34}S_{42}} - {\Gamma_{4}S_{32}S_{44}}} \right)}\end{matrix}}{S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}}.}}}$

Preferably, the scattering parameters S_(xy) are determined by measuringthe values

b₁/a₁,

b₃/a₃,

b₄/a₄,

b₃/a₁ or

b₁/a₃,

b₄/a₁ or

b₁/a₄,

b₄/a₃ or

b₃/a₄ at the measuring ports S1, S3, S4 of the calibration device, whereone or more measuring standards such as Match M, Open O and/or Short Swith the known reflection coefficients Γ_(M), Γ_(O), Γ_(S) are connectedas devices under test in the calibration plane S2, where a₁, a₃, a₄ arethe wave quantities running in at the respective measuring ports S1, S3,S4 and b₁, b₃, b₄ are the wave quantities running out at the respectivemeasuring ports S1, S3, S4.

Alternatively, determination is equally possible using other measuringstandards. In other words, for the individual measuring standards, wherenecessary the wave quantities running in and out at the three measuringports of the calibration apparatus are measured on a frequency-dependentbasis, whereby all 16 parameters of the scattering matrix S of thefour-port can be determined from these measured values by means of thefollowing equations.

S₁₁ = i₀₀ S₂₁ = i₁₀ S₁₂ = i₀₁ S₂₂ = i₁₁${S_{13} = {S_{31} = {{\overset{\sim}{S}}_{31} - {\frac{\Gamma_{DUT}i_{10}}{1 - {\Gamma_{DUT}i_{11}}}S_{32}}}}},{S_{14} = {S_{41} = {{{\overset{\sim}{S}}_{41} - {\frac{\Gamma_{DUT}i_{10}}{1 - {\Gamma_{DUT}i_{11}}}{S_{42}.S_{23}}}} = {S_{32} = \frac{{- \left( {e_{11} - i_{11}} \right)}\left( {{\Gamma_{DUT}i_{11}} - 1} \right){\overset{\sim}{S}}_{31}}{\left( {{e_{11}\Gamma_{DUT}} - 1} \right)i_{10}}}}}},{S_{24} = {S_{42} = \frac{\left( {{\Gamma_{DUT}i_{11}} - 1} \right)\left( {{e_{01}i_{10}} + {\left( {i_{11} - e_{11}} \right){\overset{\sim}{S}}_{41}}} \right)}{\left( {{e_{11}\Gamma_{DUT}} - 1} \right)i_{10}}}},{S_{33} = {{{\overset{\sim}{S}}_{33} - {\frac{\Gamma_{DUT}S_{23}}{1 - {\Gamma_{DUT}i_{11}}} \cdot {S_{32}.S_{43}}}} = {S_{34} = {{{\overset{\sim}{S}}_{34} - {\frac{\Gamma_{DUT}S_{24}}{1 - {\Gamma_{DUT}i_{11}}} \cdot {S_{32}.S_{44}}}} = {{\overset{\sim}{S}}_{44} - {\frac{\Gamma_{DUT}S_{24}}{1 - {\Gamma_{DUT}i_{11}}} \cdot S_{42}}}}}}},$

where

-   -   Γ_(DUT) is the known reflection coefficient of the calibration        standard used;    -   are the b_(x)/a_(y) measurable at the measuring ports S1, S3,        S4; and

${i_{00} = {\overset{\sim}{S}}_{11,M}},{e_{00} = \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}},{{i_{10}i_{01}} = \frac{\left( {\Gamma_{O} - \Gamma_{S}} \right)\left( {{\overset{\sim}{S}}_{11,O} - {\overset{\sim}{S}}_{11,M}} \right)\left( {{\overset{\sim}{S}}_{11,S} - {\overset{\sim}{S}}_{11,M}} \right)}{\Gamma_{O}{\Gamma_{S}\left( {{\overset{\sim}{S}}_{11,O} - {\overset{\sim}{S}}_{11,S}} \right)}}},{{e_{10}e_{01}} = \frac{\left( {\Gamma_{O} - \Gamma_{S}} \right)\left( {\frac{{\overset{\sim}{S}}_{41,O}}{{\overset{\sim}{S}}_{31,O}} - \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}} \right)\left( {\frac{{\overset{\sim}{S}}_{41,S}}{{\overset{\sim}{S}}_{31,S}} - \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{21,M}}} \right)}{\Gamma_{O}{\Gamma_{S}\left( {\frac{{\overset{\sim}{S}}_{41,O}}{{\overset{\sim}{S}}_{31,O}} - \frac{{\overset{\sim}{S}}_{41,S}}{{\overset{\sim}{S}}_{31S}}} \right)}}},{i_{11} = \frac{{\Gamma_{S}\left( {{\overset{\sim}{S}}_{11,O} - {\overset{\sim}{S}}_{11,M}} \right)} - {\Gamma_{O}\left( {{\overset{\sim}{S}}_{11,S} - {\overset{\sim}{S}}_{11,M}} \right)}}{\Gamma_{O}{\Gamma_{S}\left( {{\overset{\sim}{S}}_{11,O} - {\overset{\sim}{S}}_{11,S}} \right)}}},{e_{11} = \frac{{\Gamma_{S}\left( {\frac{{\overset{\sim}{S}}_{41,O}}{{\overset{\sim}{S}}_{31,O}} - \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}} \right)} - {\Gamma_{O}\left( {\frac{{\overset{\sim}{S}}_{41,S}}{{\overset{\sim}{S}}_{31,S}} - \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}} \right)}}{\Gamma_{O}{\Gamma_{S}\left( {\frac{{\overset{\sim}{S}}_{41,O}}{{\overset{\sim}{S}}_{31,O}} - \frac{{\overset{\sim}{S}}_{41,S}}{{\overset{\sim}{S}}_{31,S}}} \right)}}},,$

where Γ_(O), Γ_(S), Γ_(M) are the known reflection coefficients of thecalibration standards Open, Short and Match, and

are the b_(x)/a_(y) measurable at the measuring ports S1, S3, S4 withconnected calibration standard K.

These equations are simply to be understood as an exemplary means ofdetermining the scattering parameters S_(xy) of the four-port.Alternative means of calculation can also be used. However, the abovemethod has the advantage that the entries of the error matrices E and Iwhich have in any case already been determined (see documentWO2013/143650 A1) can be used, without additional measurements needingto be carried out and/or other measuring standards connected. Theskilled person will for example recognise that the S_(xy) can also bedetermined without reference to the e_(xy) and/or the i_(xy), solelyfrom the directly measurable

.

A particularly simple measuring setup using economical electroniccomponents is achieved in that the signal values v₃(t) and v₄(t) are ineach case an electric voltage.

A particularly simple and functionally reliable measuring setup isachieved in that an oscilloscope is used as time domain measuringdevice, for example a digital oscilloscope, which can be used for time-and value-range quantisation of the signal.

A particularly fast and at the same time precise transformation betweenfrequency domain and time domain which can be carried out withoutcomplex calculation is achieved in that the first mathematical operationis an FFT (Fast Fourier Transform) and the second mathematical operationan inverse FFT (IFFT—Inverse Fast Fourier Transform).

The transformation of the measured signal components v₃(t) and v₄(t)from the time domain into the frequency domain can for example becarried out according to the following calculation steps:

{V₃(l ⋅ Δ f)} = FFT{v₃(k ⋅ Δ t)}{V₄(l ⋅ Δ f)} = FFT{v₄(k ⋅ Δ t)} where  k = 0, 1, …  , N − 1${{{and}\mspace{14mu} l} = 0},1,\ldots \mspace{14mu},\frac{N - 1}{2},$

where N is a number of data points, Δf a frequency increment withΔf=2f_(max)/(N−1), Δt a time increment, where f_(max) designates themaximum frequency for which calibration data are available. Since themeasured voltages are real values, and therefore the resulting Fourierspectrum can be expected to be symmetrical around f=0, it is sufficientto consider the spectral components for f≧0.

The wave quantities b₃ and b₄ are preferably determined from thevoltages V₃ and V₄ as follows:

$b_{3} = \frac{V_{3}}{\left( {1 + \Gamma_{3}} \right)\sqrt{Z_{D}}}$${b_{4} = \frac{V_{4}}{\left( {1 + \Gamma_{4}} \right)\sqrt{Z_{D}}}},$

where Z₀ designates the impedance in relation to which the reflectioncoefficients Γ₃, Γ₄ were determined. Usually, Z₀=50Ω.

The absolute wave quantities a₂, b₂ in the calibration plane aredetermined from the wave quantities b₃, b₄, with the aid of thecalibration parameters (e_(00,r)(Γ₃, Γ₄), e_(01,r)(Γ₃, Γ₄), e_(10,r)(Γ₃,Γ₄), e_(11,r)(Γ₃, Γ₄)), through corresponding resolution of the equationsystem

${\begin{pmatrix}b_{4} \\b_{2}\end{pmatrix} = {\begin{pmatrix}e_{00,r} & e_{01,r} \\e_{10,r} & e_{11,r}\end{pmatrix}\begin{pmatrix}b_{3} \\a_{2}\end{pmatrix}}},$

and the voltage V₂ (f) and the current I₂(f) in the calibration planeare calculated from these by means of the following calculation steps:

$V_{2} = {\sqrt{Z_{1}}\left( {a_{2} + b_{2}} \right)}$${I_{2} = {\sqrt{\frac{1}{Z_{1\;}}}\left( {a_{2} - b_{2}} \right)}},$

where Z₁ designates the system impedance in the calibration plane.

The transformation of the voltage V₂(f) and of the current I₂(f) fromthe frequency domain back into the time domain can for example becarried out according to the following calculation steps:

{u(k ⋅ Δ t)} = IFFT{V₂(l ⋅ Δ f)}, {i(k ⋅ Δ t)} = IFFT{I₂(l ⋅ Δ f)}where  k = 0, 1, …  , N − 1${{{and}\mspace{14mu} l} = 0},1,\ldots \mspace{14mu},{\frac{N - 1}{2}.}$

Here too, it was possible to exploit the fact that the resulting voltageand the resulting current are real values, so that only the frequencycomponents f>0 are required as input values for the IFFT.

The desired measured values u(t) and i(t) in the calibration plane areobtained.

A vectorial network analyser (VNA or vectorial NWA) with at least threemeasuring ports is preferably used as a calibration device.

Once the calibration parameters used in the method according to theinvention have been determined in the calibration step, the voltage u(t)and/or the current i(t) in the calibration plane can then be determinedin the measuring step in that the first measuring output of thedirectional coupler and the second measuring output of the directionalcoupler are isolated from the calibration device and connected with themeasuring inputs of the time domain measuring device, while the first RFsignal is fed via the signal input of the directional coupler.

An arrangement for carrying out the calibration step of the methodaccording to the invention is represented schematically in FIG. 1. Thisarrangement features a directional coupler 18 with a signal input 19which is connected via an input cable 10 with a first measuring port S128 of a calibration device 26 (vectorial network analyser NWA). Acalibration plane 14 is connected with the signal output of thedirectional coupler. The calibration plane 14 is designed such that adevice under test (DUT) 16 can be connected electrically to thecalibration plane 14. This DUT 16 is for example a calibration standard,an electronic circuit which is to be tested or an electronic component.A component of a first RF signal, which runs within the directionalcoupler 18 from the signal input 19 in the direction of the calibrationplane 14, and a component of a second RF signal, which runs within thedirectional coupler 18 from the calibration plane 14 in the direction ofthe signal input 19, are decoupled by means of the directional coupler18 with two measuring outputs 20, 22. The first signal output 20 of thedirectional coupler 18 is connected with a second measuring port S3 30of the NWA, and the second signal output 22 of the directional coupler18 is connected with a third measuring port S4 32 of the NWA. Suitablefor use as a directional coupler 18 is any component which possessesdirectionality, i.e., which makes it possible to distinguish thecomponent of the first RF signal and the component of the second RFsignal.

A signal is input via the first measuring port 28. The calibration plane14, which is connected with the signal output of the coupler 18, isrepresented by the port S2 of the four-port S with the ports S1 to S4which is to be measured, since the device under test 16 which is to bemeasured is connected at this point. For this reason the calibrationplane is also referred to as port S2 in the following description. Thefour-port S which is to be measured, which in the present case is usedboth in the calibration step and in the measuring step, thussubstantially comprises the directional coupler 18 together with inputcables.

The four-port between the three measuring ports S1, S3 and S4 (28, 30,32) of the NWA and the calibration plane S2 14 is representedschematically in FIG. 4a . This four-port can be broken down into two(error) two-ports, which are represented schematically in FIGS. 4b and4c and as signal flow diagrams in FIGS. 3a and 3b and which can bedescribed through the two error matrices I and E. The two-port with theentries

$\quad{\begin{pmatrix}i_{00} & i_{01} \\i_{10} & i_{11}\end{pmatrix},}$

also referred to as error coefficients, is located between the measuringport S1 28 of the NWA and the calibration plane S2 14; the two-port withthe error coefficients

$\quad\begin{pmatrix}e_{00} & e_{01} \\e_{10} & e_{11}\end{pmatrix}$

is located between, on the one hand, the measuring ports S3 30 and S4 32of the NWA 26, to which the measuring outputs 20, 22 of the coupler 18are connected, and on the other hand the calibration plane S2 14.

Firstly, through the calibration, the four entries e_(xy) of the(uncorrected) error matrix E are to be determined, which express therelationship between the wave quantities a₂ and b₂ in the calibrationplane and the decoupled wave quantities b₃ and b₄ decoupled through thedirectional coupler 18 for the four-port S which is terminated in areflection-free manner at the measuring ports S3 and S4. Consequently,the NWA 26 is terminated in a reflection-free manner at its measuringports. Then, with the aid of the entries e_(xy), the (corrected)calibration parameters

$\quad\begin{pmatrix}e_{00,r} & e_{01,r} \\e_{10,r} & e_{11,r}\end{pmatrix}$

of the (corrected) error matrix E_(r) are determined, which are not onlyfrequency-dependent, but take into consideration the measuring inputs ofthe time domain measuring device 34 which are also not terminated in areflection-free manner.

In the measurement which is then to be carried out in the time domain,only the values v₃(t) and v₄(t) are determined, and from these the wavequantities b₃ and b₄ are determined, from which wave quantities as wellas voltage u(t) and current i(t) in the calibration plane 14 will thenbe derived.

For the two error two-ports, which are described through the errormatrices I and E, the following relationship can be derived from thesignal flow diagram in FIG. 3 using the reflection coefficients Γ in thecalibration plane:

$\begin{matrix}{{{\overset{\sim}{S}}_{11} = {\frac{b_{1}}{a_{1}} = {i_{00} + \frac{i_{10}{i_{01} \cdot \Gamma}}{1 - {i_{11} \cdot \Gamma}}}}},{\frac{{\overset{\sim}{S}}_{41}}{{\overset{\sim}{S}}_{31}} = {\frac{b_{4}}{b_{3}} = {e_{00} + {\frac{e_{10}{e_{01} \cdot \Gamma}}{1 - {e_{11} \cdot \Gamma}}.}}}}} & (1)\end{matrix}$

hereby designates the scattering parameters which can be measured by theNWA 26. If three calibration standards with different known reflectioncoefficients Γ_(k) are connected in the calibration plane, then linearequation systems can in each case be derived from these equations inorder to determine the error coefficients e₀₀, i₀₀, e₁₁, i₁₁, e₁₀e₀₁,i₁₀i₀₁. If one uses as calibration standards an Open (O) with thereflection coefficient Γ_(O), a Short (S) with the reflectioncoefficient Γ_(S) and a reflection-free termination (Match, M=0) withthe reflection coefficient Γ_(M)=0, then one obtains the known OSMcalibration:

$\begin{matrix}{{i_{00} = {\overset{\sim}{S}}_{11,M}},{e_{00} = \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}},{{i_{10}i_{01}} = \frac{\left( {\Gamma_{O} - \Gamma_{S}} \right)\left( {{\overset{\sim}{S}}_{11,O} - {\overset{\sim}{S}}_{11,M}} \right)\left( {{\overset{\sim}{S}}_{11,S} - {\overset{\sim}{S}}_{11,M}} \right)}{\Gamma_{O}{\Gamma_{S}\left( {{\overset{\sim}{S}}_{11,O} - {\overset{\sim}{S}}_{11,S}} \right)}}},{{e_{10}e_{01}} = \frac{\left( {\Gamma_{O} - \Gamma_{S}} \right)\left( {\frac{{\overset{\sim}{S}}_{41,O}}{{\overset{\sim}{S}}_{31,O}} - \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}} \right)\left( {\frac{{\overset{\sim}{S}}_{41,S}}{{\overset{\sim}{S}}_{31,S}} - \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}} \right)}{\Gamma_{O}{\Gamma_{S}\left( {\frac{{\overset{\sim}{S}}_{41,O}}{{\overset{\sim}{S}}_{31,O}} - \frac{{\overset{\sim}{S}}_{41,S}}{{\overset{\sim}{S}}_{31,S}}} \right)}}},{i_{11} = \frac{{\Gamma_{S}\left( {{\overset{\sim}{S}}_{11,O} - {\overset{\sim}{S}}_{11,M}} \right)} - {\Gamma_{O}\left( {{\overset{\sim}{S}}_{11,S} - {\overset{\sim}{S}}_{11,M}} \right)}}{\Gamma_{O}{\Gamma_{S}\left( {{\overset{\sim}{S}}_{11,O} - {\overset{\sim}{S}}_{11,S}} \right)}}},{e_{11} = \frac{{\Gamma_{S}\left( {\frac{{\overset{\sim}{S}}_{41,O}}{{\overset{\sim}{S}}_{31,O}} - \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}} \right)} - {\Gamma_{O}\left( {\frac{{\overset{\sim}{S}}_{41,S}}{{\overset{\sim}{S}}_{31,S}} - \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}} \right)}}{\Gamma_{O}{\Gamma_{S}\left( {\frac{{\overset{\sim}{S}}_{41,O}}{{\overset{\sim}{S}}_{31,O}} - \frac{{\overset{\sim}{S}}_{41,S}}{{\overset{\sim}{S}}_{31,S}}} \right)}}},} & (2)\end{matrix}$

where the

designate the scattering parameters measured with the calibrationstandard K.

The knowledge of these terms is sufficient in order to determine thereflection coefficient Γ_(DUT)=a₂/b₂ of a device under test (“DUT”) inthe calibration plane from the relationship between the measured wavequantities b₄/b₃. For this purpose:

$\begin{matrix}{\Gamma_{DUT} = \frac{\frac{b_{4}}{b_{3}} - e_{00}}{{e_{10} \cdot e_{01}} + {e_{11}\left( {\frac{b_{4}}{b_{3}} - e_{00}} \right)}}} & (3)\end{matrix}$

However, in order to determine the absolute wave quantities a₂ and b₂from b₃ and b₄ it is necessary to resolve the product e₁₀e₀₁ into itsfactors. For this purpose, the calibration is extended in the following.It should firstly be noted that the error matrix E obtained through themathematical operation of the four-port-two-port reduction does notdescribe a reciprocal two-port, i.e. e₁₀≠e₀₁. In contrast, the errormatrix I describes the relationship between the measuring port S1 of theNWA 26 and the calibration plane 14, S2 and can thus be assumed to bereciprocal. Thus:

i ₁₀ =i ₀₁=±√{square root over (i ₁₀ ·i ₀₁)}  (4)

The decision regarding the correct sign in equation (4) is equivalent tothe correct determination of the phase of i₁₀ from two possibilities. Todo this, one proceeds as follows: the phase at a frequency point must beknown with adequate precision in order to decide on the sign. This canfor example be done through an estimation of the electric length of thesetup between the measuring port S1 of the NWA 26 and the calibrationplane S2, 14. It is also assumed that the phase changes by less than 90°between two adjacent frequency points. The correct phase of i₁₀ can thusalso be determined for all frequency points. The following relationshipsfor b₂ can be derived from the graphs in FIGS. 3a and 3b :

$\begin{matrix}{{b_{2} = \frac{i_{10}a_{1}}{1 - {i_{11}\Gamma_{DUT}}}}{b_{2} = \frac{e_{10}b_{3}}{1 - {e_{11}\Gamma_{DUT}}}}} & (5)\end{matrix}$

Since both equations describe the same wave quantity, this means that

$\begin{matrix}{e_{10} = {i_{10} \cdot \frac{a_{1}}{b_{3}} \cdot \frac{1 - {e_{11}\Gamma_{DUT}}}{1 - {i_{11}\Gamma_{DUT}}}}} & (6)\end{matrix}$

In this case,

${\frac{a_{1}}{b_{3}} = {\overset{\sim}{S}}_{31}^{- 1}},$

so that e₁₀ can be determined individually and, from this, using (2),also e₀₁. Using (3), (5) and the relationship

$\begin{matrix}{a_{2} = \frac{b_{4} - {e_{00}b_{3}}}{e_{01}}} & (7)\end{matrix}$

derived from the signal flow graphs according to FIG. 3, afterdetermining the four coefficients of the error matrix E from measured b₃and b₄, the absolute wave quantities a₂ and b₂ in the calibration plane14 can now be determined for a time domain measuring device withreflection-free termination.

However, time domain measuring devices 34 such as oscilloscopes aregenerally not terminated in a perfectly reflection-free manner. Rather,they can display reflection coefficients Γ₃≠0 and/or Γ₄≠0 at theirmeasuring inputs 36, 38, at which the signal components 72, 74 run in inthe measuring step shown in FIG. 2. The entries e_(xy) of the(uncorrected) error matrix E described above were determined on theassumption of a reflection-free termination of the time domain measuringdevice 34, so that no exact voltages and/or currents in the calibrationplane can be determined if a time domain measuring device 34 with Γ₃≠0and/or Γ₄≠0 is used.

In the following description, the procedure for determining thecorrected calibration parameters

$\quad\begin{pmatrix}e_{00,r} & e_{01,r} \\e_{10,r} & e_{11,r}\end{pmatrix}$

of the corrected error matrix E_(r) with reference to the aforementioned(uncorrected) error matrix E is described, whereby in the methodaccording to the invention the corrected calibration parameters e_(xy,r)are used.

This example involves the four-port as shown in FIG. 3a , describedthrough its scattering matrix S. If this four-port represents thedirectional coupler 18 together with input cables, then the signal input19 of the directional coupler 18 is connected with the measuring port S128 of the network analyser 26 during the calibration, and during themeasurement is for example connected with a signal source 24. Themeasuring outputs 20 and 22 of the four-port/directional coupler 18 areconnected with the measuring ports S3 and S4 of the NWA 26 during thecalibration and are connected with the measuring inputs 36, 38 of thetime domain measuring device 34 during the measurement. The calibrationstandards are connected with port S2 of the four-port, the calibrationplane 14, during the calibration and the device under test 16 which isto be measured is connected with this during the measurement. It istherefore the object of the calibration procedure to determine therelationship between the measurable wave quantities b₃ and b₄ and thewave quantities a₂ and b₂ in the calibration plane 14. This relationshipcan, as explained above, be represented as a—not physicallypresent—two-port with the error matrix E according to FIG. 3a :

$\begin{matrix}{\begin{pmatrix}b_{4} \\b_{2}\end{pmatrix} = {\begin{pmatrix}e_{00} & e_{01} \\e_{10} & e_{11}\end{pmatrix}\begin{pmatrix}b_{3} \\a_{2}\end{pmatrix}}} & (8)\end{matrix}$

As already shown above, all four scattering parameters or error termse_(xy) of this error two-port can be determined through the calibration,without explicit knowledge of the scattering matrix S of the underlyingfour-port. Nonetheless, the relationship between the two matrices E andS will be derived in the following. Under the assumption that reflectioncoefficients Γ₃ and Γ₄ occur at the measuring points S3 and S4, withwhich the measuring outputs of the directional coupler are connected,the following six equations can be arrived at:

$\begin{matrix}{{\begin{pmatrix}b_{1} \\b_{2} \\b_{3} \\b_{4}\end{pmatrix} = {{S\begin{pmatrix}a_{1} \\a_{2} \\a_{3} \\a_{4}\end{pmatrix}} = {\begin{pmatrix}S_{11} & S_{12} & S_{13} & S_{14} \\S_{21} & S_{22} & S_{23} & S_{24} \\S_{31} & S_{32} & S_{33} & S_{34} \\S_{41} & S_{42} & S_{43} & S_{44}\end{pmatrix}\begin{pmatrix}a_{1} \\a_{2} \\a_{3} \\a_{4}\end{pmatrix}}}},{a_{3} = {\Gamma_{3}b_{3}}},{a_{4} = {\Gamma_{4}{b_{4}.}}}} & (9)\end{matrix}$

It had been assumed above that Γ₃=Γ₄=0 also applies to the time domainmeasuring device 34. Under this assumption, the (uncorrected) errormatrix E is derived from equation (9) as follows (four-port-two-portreduction):

$\begin{matrix}{\begin{pmatrix}b_{4} \\b_{2}\end{pmatrix} = {{\begin{pmatrix}e_{00} & e_{01} \\e_{10} & e_{11}\end{pmatrix}\begin{pmatrix}b_{3} \\a_{2}\end{pmatrix}} = {\begin{pmatrix}\frac{S_{41}}{S_{31}} & \frac{{S_{31}S_{42}} - {S_{32}S_{41}}}{S_{31}} \\\frac{S_{21}}{S_{31}} & \frac{{S_{22}S_{31}} - {S_{21}S_{32}}}{S_{31}}\end{pmatrix}\begin{pmatrix}b_{3} \\a_{2}\end{pmatrix}}}} & (10)\end{matrix}$

If, in contrast, as when carrying out the measurement with anoscilloscope for example, it is assumed that Γ₃≠0 and/or Γ₄≠0 then thefollowing equations result for the error terms e_(xy,r) which take intoconsideration these reflection coefficients:

$\begin{matrix}{\mspace{79mu} {{e_{00,r} = {\xi \cdot \left\lbrack {S_{41} - {\Gamma_{3}S_{33}S_{41}} + {\Gamma_{3}S_{31}S_{43}}} \right\rbrack}},\mspace{20mu} {e_{01,r} = {\xi \cdot \left\lbrack {{S_{31}S_{42}} - {S_{32}S_{41}}} \right\rbrack}},{e_{10,r} = {\xi \cdot \left\lbrack {{\Gamma_{4}S_{24}S_{41}} + {\Gamma_{3}\left( {{\Gamma_{4}{S_{24}\left( {{S_{31}S_{43}} - {S_{33}S_{41}}} \right)}} + {S_{23}\left( {S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}{xS}_{31}S_{44}}} \right)}} \right)} + {S_{21}\left( {1 - {\Gamma_{4}S_{44}} - {\Gamma_{3}\left( {S_{33} + {\Gamma_{4}S_{34}S_{43}} - {\Gamma_{4}S_{33}S_{44}}} \right)}} \right)}} \right\rbrack}},{e_{11,r} = {\xi \cdot \left\lbrack {{{\Gamma_{4}{S_{24}\left( {{{- S_{32}}S_{41}} + {S_{31}S_{42}}} \right)}} + {{S_{22}\left( {S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}} - {S_{21}\left( {S_{32} + {\Gamma_{4}S_{34}S_{42}} - {\Gamma_{4}S_{32}S_{44}}} \right)}} \right\rbrack}\mspace{20mu} {mit}\mspace{20mu} \xi}} = {\frac{1}{S_{31} + {\Gamma_{4}S_{34}S_{41}} + {\Gamma_{4}S_{31}S_{44}}}.}} \right.}}}} & (11)\end{matrix}$

If the scattering parameters S_(xy) of the four-port are known, thenconsequently calibration parameters can be calculated which take intoaccount any known mismatches Γ_(x) at the measuring inputs of the timedomain measuring device 34. In the following section it will beexplained how these scattering parameters S_(xy) can be obtained duringthe calibration of the directional coupler 18 without additionalcalibration steps or standards.

The scattering parameters of the four-port according to FIG. 4a cannotbe determined through a direct measurement since the calibrationplane—which is also the port S2 of the four-port—is not connected withthe NWA. If, however, a reciprocal four-port is used, these parameterscan nonetheless be determined. Use can be made of the fact thatcalibration standards with known reflection coefficient Γ_(DUT) in thecalibration plane are present during the calibration. The calibrationstandard with which the following calculation is carried out can therebybe chosen arbitrarily, for example Γ_(DUT)=Γ_(O), Γ_(DUT)=Γ_(S) orΓ_(DUT)=Γ_(M) apply selectively. If it is also assumed that apart fromthe calibration plane S2 all other ports of the four-port S1, S3, S4 areterminated in a reflection-free manner during the calibration, then onbeing input via the port S1:

$\begin{matrix}{a_{2} = {{\Gamma_{DUT}b_{2}} = {{\Gamma_{DUT}\left( {{S_{22}a_{2}} + {S_{21}a_{1}}} \right)} = {\frac{\Gamma_{DUT}S_{21}a_{1}}{1 - {\Gamma_{DUT}S_{22}}}.}}}} & (12)\end{matrix}$

If the NWA is now used for example to measure b₃/a₁, then these are notthe scattering parameters S₃₁ of the four-port, since the necessarycondition a₂=0 is not generally fulfilled. Therefore, the valuesmeasured by the NWA are designated as

$= \frac{b_{y}}{a_{x}}$

in order to distinguish them from the scattering parameters S_(xy) ofthe four-port. In order to be able to carry out the subsequentcorrection of the error terms by known reflection coefficients, asdescribed above, the “true” scattering parameters of the four-portS_(xy) must be determined from the measurements by the NWA

$= {\frac{b_{y}}{a_{x}}.}$

The already-determined error coefficients e_(xy) and i_(xy) are alsoused. If the definition of the error matrix I according to FIG. 3b iscompared with the four-port S, then one obtains

S ₁₁ =i ₀₀

S ₂₁ =i ₁₀

S ₁₂ =i ₀₁

S ₂₂ =i ₁₁  (13)

Also, with input at port S1 (a₃=a₄=0), with the equations (12), (13) anda comparison of the e_(xy) according to equation (2) and equation (10)one obtains:

$\begin{matrix}{{S_{23} = {S_{32} = \frac{{- \left( {e_{11} - i_{11}} \right)}\left( {{\Gamma_{DUT}i_{11}} - 1} \right){\overset{\sim}{S}}_{31}}{\left( {{e_{11}\Gamma_{DUT}} - 1} \right)i_{10}}}},{S_{24} = {S_{42} = \frac{\left( {{\Gamma_{DUT}i_{11}} - 1} \right)\left( {{e_{01}i_{10}} + {\left( {i_{11} - e_{11}} \right){\overset{\sim}{S}}_{41}}} \right)}{\left( {{e_{11}\Gamma_{DUT}} - 1} \right)i_{10}}}},{S_{13} = {S_{31} = {{\overset{\sim}{S}}_{31} - {\frac{\Gamma_{DUT}i_{10}}{1 - {\Gamma_{DUT}i_{11}}}S_{32}}}}},{S_{14} = {S_{41} = {{\overset{\sim}{S}}_{41} - {\frac{\Gamma_{DUT}i_{10}}{1 - {\Gamma_{DUT}i_{11}}}{S_{42}.}}}}}} & (14)\end{matrix}$

Under the assumption that the four-port is now fed in via the port S3and the ports S1, S4 are terminated in a reflection-free manner, it canalso be stated that:

$\begin{matrix}{{b_{3} = {{S_{33}a_{3}} + {S_{32}a_{2}}}},{a_{2} = {{\Gamma_{DUT}b_{2}} = {{\Gamma_{DUT}\left( {{S_{22}a_{2}} + {S_{23}a_{3}}} \right)} = {\frac{\Gamma_{DUT}S_{23}}{1 - {\Gamma_{DUT}S_{22\;}}} \cdot a_{3}}}}},{{\overset{\sim}{S}}_{33} = {\frac{b_{3}}{a_{3}}{_{{a\; 2} \neq 0}{{= {S_{33} + {\frac{\Gamma_{DUT}S_{23}}{1 - {\Gamma_{DUT}S_{22}}} \cdot S_{32}}}},{S_{33} = {{\overset{\sim}{S}}_{33} - {\frac{\Gamma_{DUT}S_{23}}{1 - {\Gamma_{DUT}i_{11}}} \cdot {S_{32}.}}}}}}}}} & (15)\end{matrix}$

S₄₄ and S₃₄ of the four-port can also be derived if, analogously, aninput via the port S4 is considered:

$\begin{matrix}{{S_{44} = {{\overset{\sim}{S}}_{44} - {\frac{\Gamma_{DUT}S_{24}}{1 - {\Gamma_{DUT}i_{11}}} \cdot S_{42}}}},{S_{43} = {S_{34} = {{\overset{\sim}{S}}_{34} - {\frac{\Gamma_{DUT}S_{24}}{1 - {\Gamma_{DUT}i_{11}}} \cdot {S_{32}.}}}}}} & (16)\end{matrix}$

Thus, all 16 scattering parameters S_(xy) are determined during thecourse of the calibration, without there needing to be a directconnection between the calibration plane S2 and the NWA 26. Using theequations (11), the corrected calibration parameters e_(xy,r), whichtake into account any known mismatches during the measurement, can bedetermined in this way.

The measuring step for determining the electric voltage u(t) and theelectric current i(t) of the RF signal in the calibration plane (14) inthe method according to the invention is described in the following.

FIG. 2 shows the setup for measuring the voltage u(t) and the currenti(t) in the calibration plane 14 from the measured values of anoscilloscope or another time domain measuring device 34 v₃(t) 72 andv₄(t) 74 in a measuring plane. The measuring inputs 36, 38 of the timedomain measuring device 34 are hereby connected with the measuringoutputs 20 and 22 of the directional coupler 18, and any desired signalsource 24 b is connected with the signal input 19 of the directionalcoupler 18 via a, possibly modified, input cable 10 b.

The use of the calibration parameters e_(xy,r) is explained in thefollowing. It should be emphasised that only the marked part of thesetup (enclosed in a broken line) needs to remain unchanged incomparison with the calibration. This substantially involves thedirectional coupler 18 as far as the calibration plane 14 as well as thecables which connect the directional coupler 18 with the time domainmeasuring device 34. In contrast, changes—also in the characteristicimpedance—to the other elements, for example the source and the load,have no influence on the measurement. It is assumed that the setupbetween the calibration plane 14 and the measuring inputs 36, 38 and themeasuring outputs 20, 22 of the directional coupler 18 does not changein comparison with the calibration according to FIG. 1 if the obtainedcalibration coefficients are to remain valid. In contrast, changes tothe signal source 24 b and its input cable 10 b to the directionalcoupler have no influence on the calibration.

In order to use the error matrix E_(r) defined in the frequency domain,the voltages v₃(t) and v₄(t) recorded by the oscilloscope in the timedomain are transformed into the corresponding values V₃(f) and V₄(f) inthe frequency domain. In the following representation, the fast Fouriertransform (FFT) is used for this purpose. Alternatively, in order to beable to process the large quantities of data which occur duringmeasurements carried out with a high sampling rate in blocks with anadjustable time and frequency resolution, the short-time Fouriertransform (STFT) can be used. Since, as a result of the measurement inthe time domain, the phase information is inherently maintained betweenall spectral components, this setup is not limited to the measurement ofmonofrequency or periodic signals.

The measured voltages are represented—possibly through interpolation—astime-discrete vectors {v₃ (k·Δt)} or {v₄(k·Δt)} with a time incrementΔt=0.5/f_(max), where f_(max) designates the maximum frequency for whichcalibration data are available and k=0, 1, . . . , N−1 is a runningindex over all N data points. These vectors are transformed into thefrequency domain with the aid of the fast Fourier transform (FFT) andare then designated as V₃(f) and V₄(f):

$\begin{matrix}{\left\{ {V_{3}\left( {{l \cdot \Delta}\; f} \right)} \right\} = {{FFT}\left\{ {v_{3}\left( {{k \cdot \Delta}\; t} \right)} \right\}}} & (17) \\{{\left\{ {V_{4}\left( {{l \cdot \Delta}\; f} \right)} \right\} = {{FFT}\left\{ {v_{4}\left( {{k \cdot \Delta}\; t} \right)} \right\}}}{{{{where}\mspace{14mu} k} = 0},1,\ldots \mspace{14mu},{N - 1}}{{{{and}\mspace{14mu} l} = 0},1,\ldots \mspace{14mu},{\frac{N - 1}{2}.}}} & (18)\end{matrix}$

Since the measured voltages are real values, it is sufficient toconsider the spectral components for f≧0. This leads to a frequencyincrement of Δf=2f_(max)/(N−1). The calibration coefficients e_(xy,r)are brought into the same frequency pattern through interpolation.

With known reflection coefficients Γ₃, Γ₄ of the measuring inputs 36, 38of the time domain measuring device 34, the following relationshipbetween the voltages V₃ and V₄ and the wave quantities b₃ and b₄ resultsfor each frequency point:

$\begin{matrix}{b_{3} = \frac{V_{3}}{\left( {1 + \Gamma_{3}} \right)\sqrt{Z_{D}}}} & (19) \\{{b_{4} = \frac{V_{4}}{\left( {1 + \Gamma_{4}} \right)\sqrt{Z_{D}}}},} & (20)\end{matrix}$

where Z₀ designates the impedance in relation to which the reflectioncoefficients Γ₃, Γ₄ were determined. Since it was assumed during thecalibration that Γ₃=Γ₄=0, the system impedance of the calibrated NWAdetermines the impedance Z₀. Usually, Z₀ is therefore 50Ω.

The absolute wave quantities a₂, b₂ in the calibration plane aredetermined from these wave quantities with the aid of the calibrationparameters (e_(00,r)(Γ₃, Γ₄), e_(01,r)(Γ₃, Γ₄), e_(10,r)(Γ₃, Γ₄),e_(11,r)(Γ₃, Γ₄)) using the equations (3), (5) and (7), where e_(xy) isin each case replaced with e_(xy,r)(Γ₃, Γ₄), and the voltage V₂ and thecurrent I₂ in the calibration plane are derived from these:

$\begin{matrix}{V_{2} = {\sqrt{Z_{1}}\left( {a_{2} + b_{2}} \right)}} & (21) \\{{I_{2} = {\sqrt{\frac{1}{Z_{1\;}}}\left( {a_{2} - b_{2}} \right)}},} & (22)\end{matrix}$

where Z₁ designates the system impedance in the calibration plane. Thisis the impedance on which the statement of the reflection coefficientsΓ_(O,S,M) of the calibration standards was based during the calibration.This need not be the physical line impedance in the calibration plane14. Any choice of Z₁, which must, however, be consistent betweencalibration and measurement, has no influence on the measurement result.The time-discrete representation of the voltage u(t) and of the currenti(t) in the calibration plane can be obtained from V₂(f) and I₂(f) withthe aid of the inverse FFT (IFFT):

$\begin{matrix}{{\left\{ {u\left( {{k \cdot \Delta}\; f} \right)} \right\} = {{IFFT}\left\{ {V_{2}\left( {{l \cdot \Delta}\; f} \right)} \right\}}},} & (23) \\{{\left\{ {i\left( {{k \cdot \Delta}\; f} \right)} \right\} = {{IFFT}\left\{ {I_{2}\left( {{l \cdot \Delta}\; f} \right)} \right\}}}{{{{where}\mspace{14mu} k} = 0},1,\ldots \mspace{14mu},{N - 1}}{{{{and}\mspace{14mu} l} = 0},1,\ldots \mspace{14mu},{\frac{N - 1}{2}.}}} & (24)\end{matrix}$

Here too it was possible to make use of the fact that the resultingvoltage and the resulting current are real values, so that only thefrequency components f>0 are required as input values for the IFFT.

In order to verify that the voltage u(t) and the current i(t) in thecalibration plane 14 can also be determined exactly with the time domainmeasurement, with the aid of the calibration parameters e_(xy,r), withnon-ideal termination of the measuring inputs, a simulation was carriedout using the software “Agilent ADS”. A line coupler was used asdirectional coupler 18 for the measurement. The calibration step wascarried out with an ideal 50Ω system. For the error matrix

${E = \begin{pmatrix}e_{00} & e_{01} \\e_{10} & e_{11}\end{pmatrix}},$

the frequency-dependent values shown in FIG. 6a are obtained.

For the measurement, on the other hand, to achieve a 50Ω termination atthe measuring ports of the directional coupler a capacitance of 1 nF wasconnected in parallel. In the simulated measurement, this leads to thefrequency-dependent reflection coefficients Γ₃=Γ₄≠0.

In order firstly to show the correct determination of the scatteringparameters S_(xy) by means of the equations (13) to (16), the scatteringparameters are determined in a separate simulation, in which thecalibration plane is replaced by a port S2. FIG. 5 shows the perfectcorrespondence between this reference and the 16 scattering parametersof the four-port determined according to the equations. The scatteringparameters can therefore now be used to determine a corrected matrixE_(r), corrected by the mismatch, the entries of which are thecalibration parameters e_(xy,r), which are represented in FIG. 6 b.

FIGS. 7a and 7b show the voltage u(t) and the current i(t) in thecalibration plane. It can be seen that where the calibration parameterse_(xy) which are not corrected by the mismatch are used, as shown inFIG. 6a , neither the amplitude nor the form of the curves are correctlyreproduced. In contrast, both the voltage and the current are consistentwith the reference determined through simulation if the correctedcalibration parameters e_(xy,r) corrected by means of the methodaccording to the invention are used (see FIG. 6b ). It could thus beverified that, with corresponding correction through the methodaccording to the invention, a time domain measuring device withmeasuring inputs with reflection-free termination need not necessarilybe used, and that reflection coefficients Γ₃≠0 and/or Γ₄≠0 can becorrected.

While the present invention has been particularly described, inconjunction with a specific preferred embodiment, it is evident thatmany alternatives, modifications and variations will be apparent tothose skilled in the art in light of the foregoing description. It istherefore contemplated that the appended claims will embrace any suchalternatives, modifications and variations as falling within the truescope and spirit of the present invention.

1. A method for determining an electric voltage u(t) and/or an electriccurrent i(t) of a RF signal on an electric cable in a calibration planethrough measurement in the time domain using a time domain measuringdevice, wherein a device under test can be connected electrically withthe calibration plane, wherein, in a measuring step, using a directionalcoupler, a first component v₃(t) of a first RF signal which, startingout from a signal input, runs in the direction of the calibration planethrough the directional coupler, is decoupled, fed to the time domainmeasuring device at a first measuring input and measured there, and asecond component v₄(t) of a second RF signal which, starting out fromthe calibration plane, runs in the direction of the signal input throughthe directional coupler, is decoupled, fed to the time domain measuringdevice at a second measuring input and measured there, wherein thesignal components v₃(t), v₄(t) are, by a first mathematical operation,transformed into the frequency domain as wave quantities V₃(f) andV₄(f), then absolute wave quantities a₂ and b₂ in the frequency domainare determined in the calibration plane from the wave quantities V₃(f)and V₄(f) using calibration parameters (e_(00,r), e_(01,r), e_(10,r),e_(11,r)), and the determined absolute wave quantities a₂ and b₂ are, bya second mathematical operation, converted into the electric voltageu(t) and/or the electric current i(t) of the RF signal in the timedomain in the calibration plane, wherein the calibration parameters linkthe wave quantities V₃(f) and V₄(f) mathematically with the absolutewave quantities a₂ and b₂ in the calibration plane, such that the firstmeasuring input of the time domain measuring device has a reflectioncoefficient Γ₃≠0 and/or the second measuring input of the time domainmeasuring device has a reflection coefficient Γ₄≠0, and in a precedingcalibration step, the calibration parameters (e_(00,r), e_(01,r),e_(10,r), e_(11,r)) are determined, with the aid of a calibrationdevice, in relation to the frequency f and in relation to a calibrationstandard with known reflection coefficient Γ_(DUT) of at least one ofthe measuring inputs of the time domain measuring device, and the wavequantities a₂ and b₂ are determined in the measuring step from the wavequantities V₃(f) and V₄(f) using the calibration parameters(e_(00,r)(Γ₃, Γ₄), e_(01,r)(Γ₃, Γ₄), e_(10,r)(Γ₃, Γ₄), e_(11,r)(Γ₃,Γ₄)), wherein, during the calibration step the signal input of thedirectional coupler is connected with a first measuring port S1, thefirst measuring output of the directional coupler is connected with asecond measuring port S3 and the second measuring output of thedirectional coupler is connected with a third measuring port S4 of thecalibration device, wherein one or more measuring standards with knownreflection coefficients are connected to a signal output of thedirectional coupler connected with the calibration plane S2, wherein thecalibration parameters (e_(00,r), e_(01,r), e_(10,r), e_(11,r)) link thewave quantity b₃ running in at the second measuring port S3 and the wavequantity b₄ running in at the third measuring port S4 with the wavequantities b₂, a₂ running in and out in the calibration plane (14, S2)as follows: $\begin{pmatrix}b_{4} \\b_{2\;}\end{pmatrix} = {\begin{pmatrix}e_{00,r} & e_{01,r} \\e_{10,r} & e_{11,r}\end{pmatrix}\begin{pmatrix}b_{3} \\a_{2}\end{pmatrix}}$ wherein the scattering parameters S_(xy) (x=1-4, y=1-4)of the scattering matrix S of the four-port with the ports S1, S2, S3,S4, in particular of the directional coupler together with input cables,are determined with the aid of the calibration apparatus, wherein thecalibration parameters e_(00,r), e_(01,r), e_(10,r), e_(11,r) inrelation to the reflection coefficients of the time domain measuringdevice Γ₃ Γ₄ are determined from the scattering parameters S_(xy),wherein the calibration parameters are determined from the scatteringparameters as follows:$\mspace{20mu} {{e_{00,r} = \frac{S_{41} - {\Gamma_{3}S_{33}S_{41}} + {\Gamma_{3}S_{31}S_{43}}}{S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}}},\mspace{20mu} {e_{01,r} = \frac{{S_{31}S_{42}} - {S_{32}S_{41}}}{S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}}},{e_{10,r} = {\frac{{\Gamma_{4}S_{24}S_{41}} + {\Gamma_{3}\begin{pmatrix}{{\Gamma_{4}{S_{24}\left( {{S_{31}S_{43}} - {S_{33}S_{41}}} \right)}} +} \\{S_{23}\left( {S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}} \right)}\end{pmatrix}}}{S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}} + \frac{S_{21}\left( {1 - {\Gamma_{4}S_{44}} - {\Gamma_{3}\left( {S_{33} + {\Gamma_{4}S_{34}S_{43}} - {\Gamma_{4}S_{33}S_{44}}} \right)}} \right)}{{S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44\;}}}\;}}},{e_{11,r} = \frac{\begin{matrix}{{\Gamma_{4}{S_{24}\left( {{{- S_{32}}S_{41}} + {S_{31}S_{42}}} \right)}} + {S_{22}\left( {S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}} \right)} -} \\{S_{21}\left( {S_{32} + {\Gamma_{4}S_{34}S_{42}} - {\Gamma_{4}S_{32}S_{44}}} \right)}\end{matrix}}{S_{31} + {\Gamma_{4}S_{34}S_{41}} - {\Gamma_{4}S_{31}S_{44}}}}}$wherein the scattering parameters S_(xy) are determined throughmeasurement of the values b₁/a₁, b₃/a₃, b₄/a₄, b₃/a₁ or b₁/a₃, b₄/a₁ orb₁/a₄, b₄/a₃ or b₃/a₄ at the measuring ports S1, S3, S4 of thecalibration device, wherein in each case preferably the measuringstandards Match (M), Open (O), Short (S) with the known reflectioncoefficients Γ_(M), Γ_(O), Γ_(S) are connected as devices under test inthe calibration plane S2, where a₁, a₃, a₄ are wave quantities runningin at the respective measuring ports S1, S3, S4 and b₁, b₃, b₄ are wavequantities running out at the respective measuring ports S1, S3, S4, andwherein the scattering parameters S_(xy) are determined by means of thefollowing equations: S₁₁ = i₀₀ S₂₁ = i₁₀ S₁₂ = i₀₁ S₂₂ = i₁₁${S_{13} = {S_{31} = {{\overset{\sim}{S}}_{31} - {\frac{\Gamma_{DUT}i_{10}}{1 - {\Gamma_{DUT}i_{11}}}S_{32}}}}},{S_{14} = {S_{41} = {{{\overset{\sim}{S}}_{41} - {\frac{\Gamma_{DUT}i_{10}}{1 - {\Gamma_{DUT}i_{11}}}{S_{42}.S_{23}}}} = {S_{32} = \frac{{- \left( {e_{11} - i_{11}} \right)}\left( {{\Gamma_{DUT}i_{11}} - 1} \right){\overset{\sim}{S}}_{31}}{\left( {{e_{11}\Gamma_{DUT}} - 1} \right)i_{10}}}}}},{S_{24} = {S_{42} = \frac{\left( {{\Gamma_{DUT}i_{11}} - 1} \right)\left( {{e_{10}i_{10}} + {\left( {i_{11} - e_{11}} \right){\overset{\sim}{S}}_{41}}} \right)}{\left( {{e_{11}\Gamma_{DUT}} - 1} \right)i_{10}}}},{S_{33} = {{{\overset{\sim}{S}}_{33} - {\frac{\Gamma_{DUT}S_{23}}{1 - {\Gamma_{DUT}i_{11}}} \cdot {S_{32}.S_{43}}}} = {S_{34} = {{{\overset{\sim}{S}}_{34} - {\frac{\Gamma_{DUT}S_{24}}{1 - {\Gamma_{DUT}i_{11}}} \cdot {S_{32}.S_{44}}}} = {{\overset{\sim}{S}}_{44} - {\frac{\Gamma_{DUT}S_{24}}{1 - {\Gamma_{DUT}i_{11}}} \cdot S_{42}}}}}}},$where: Γ_(DUT) is the known reflection coefficient of the calibrationstandard used during the measurement:

are the b_(x)/a_(y) measurable at the measuring ports S1, S3, S4; and${i_{00} = {\overset{\sim}{S}}_{11,M}},{e_{00} = \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}},{{i_{10}i_{01}} = \frac{\left( {\Gamma_{O} - \Gamma_{S}} \right)\left( {{\overset{\sim}{S}}_{11,O} - {\overset{\sim}{S}}_{11,M}} \right)\left( {{\overset{\sim}{S}}_{11,S} - {\overset{\sim}{S}}_{11,M}} \right)}{\Gamma_{O}{\Gamma_{S}\left( {{\overset{\sim}{S}}_{11,O} - {\overset{\sim}{S}}_{11,S}} \right)}}},{{e_{10}e_{01}} = \frac{\left( {\Gamma_{O} - \Gamma_{S}} \right)\left( {\frac{{\overset{\sim}{S}}_{41,O}}{{\overset{\sim}{S}}_{31,O}} - \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}} \right)\left( {\frac{{\overset{\sim}{S}}_{41,S}}{{\overset{\sim}{S}}_{31,S}} - \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}} \right)}{\Gamma_{O}{\Gamma_{S}\left( {\frac{{\overset{\sim}{S}}_{41,O}}{{\overset{\sim}{S}}_{31,O}} - \frac{{\overset{\sim}{S}}_{41,S}}{{\overset{\sim}{S}}_{31,S}}} \right)}}},{i_{11} = \frac{{\Gamma_{S}\left( {{\overset{\sim}{S}}_{11,O} - {\overset{\sim}{S}}_{11,M}} \right)} - {\Gamma_{O}\left( {{\overset{\sim}{S}}_{11,S} - {\overset{\sim}{S}}_{11,M}} \right)}}{\Gamma_{O}{\Gamma_{S}\left( {{\overset{\sim}{S}}_{11,O} - {\overset{\sim}{S}}_{11,S}} \right)}}},{e_{11} = \frac{{\Gamma_{S}\left( {\frac{{\overset{\sim}{S}}_{41,O}}{{\overset{\sim}{S}}_{31,O}} - \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}} \right)} - {\Gamma_{O}\left( {\frac{{\overset{\sim}{S}}_{41,S}}{{\overset{\sim}{S}}_{31,S}} - \frac{{\overset{\sim}{S}}_{41,M}}{{\overset{\sim}{S}}_{31,M}}} \right)}}{\Gamma_{O}{\Gamma_{S}\left( {\frac{{\overset{\sim}{S}}_{41,O}}{{\overset{\sim}{S}}_{31,O}} - \frac{{\overset{\sim}{S}}_{41,S}}{{\overset{\sim}{S}}_{31,M}}} \right)}}},$where Γ_(O), Γ_(S), Γ_(M) are known reflection coefficients of thecalibration standards Open (O), Short (S) and Match (M), and

are the b_(x)/a_(y) measurable at the measuring ports with connectedcalibration standard K.
 2. The method of claim 1, wherein the signalcomponents v₃(t) and/or v₄(t) are an electric voltage.
 3. The method ofclaim 1 including using an oscilloscope as the time domain measuringdevice.
 4. The method of claim 1 wherein the first mathematicaloperation is an FFT (Fast Fourier Transform) and/or the secondmathematical operation is an inverse FFT (IFFT—Inverse Fast FourierTransform).
 5. The method of claim 1 wherein a vectorial networkanalyser (VNA) with at least three measuring ports is used as thecalibration device. 6-11. (canceled)
 12. The method of claim 1, whereinduring the measuring step, in order to measure the time-variable signalcomponents u(t) and i(t), the first measuring output of the directionalcoupler and the second measuring output of the directional coupler areisolated from the calibration device and connected with the measuringinputs of the time domain measuring device, while the first RF signal isfed via the signal input of the directional coupler.